Global axis system

In general, the position of a nonlinear rigid polyatomic molecule relative to a global axis system (fixed in space) can be defined by six coordinates three to define the position of the center of mass and three (usually the Euler angles a, p, y) to define the orientation of its local or molecule-fixed axis system relative to the global axis system. The local axis system is defined by the bonds within the molecule and should be chosen to reflect any symmetry because this simplifies the model potential. For example, a water molecule with C21, symme-... 

Unless there is a macroscopic feature, such as an external electric field, defining the global axis system, the coordinate system is just an abstract frame of reference, and U(R, Q.) cannot depend on the relative orientation of the two... 

In the example of H2O above we used the idea of a global axis system, X, Y, Z. This axis system is used to define the positions of the symmetry elements of the molecule and, once set, the global axis system is not moved by any operations that are carried out. This means that the symmetry elements should be considered immovable and symmetry operations only move the atoms in the molecule. This becomes especially important when molecules with more symmetry elements are considered. For example, ammonia (NH3) has a principal axis of order 3 and three vertical mirror planes, as shown in Figure 2.3. 

As before, when generating products of operations for the multiplication table, the symmetry elements are thought of as fixed in space, set by the global axis system. For example, 2 will alter the positions of the hydrogen atoms but not the location of C2 . 

The symmetry elements of a point group are defined with respect to a global axis system and so do not move under any of the operations of the group. 

Figure 2 Examples of global and local axis systems, (a) Molecular axis system for a homonuclear diatomic. Wth this system, all central multipoles with k 0 otl odd will be zero, and no S functions with k 0 oxl odd can appear in a molecule-molecule expansion of U(R, Q). The atomic multipoles Q q (all / 0 allowed) on the two atoms will be related by Qio = (-l) Qio- ( ) Local atomic axis system for a homonuclear diatomic molecule. With this definition QJq = Qio- (c) Molecular axis system for water. The nonzero atomic multipole moments for the O atom would be QoO> QlO) QzOj Qz2c = (Q22 + Qz-2)1 > QsOJ Q32c = (Qs2 + on the hydrogen atoms Qjo = Qoo. Qio = Qio> Qiic = (-Q11 + Qi-i) = -Qiic...

To handle these variables, we expand each correlation function in an angle-dependent basis set of rotational invariants [258]. Taking advantage of the fact that, in an globally isotropic system, the direction of the wavevector k does not matter, we choose k to be parallel to the -axis of the space-fixed coordinate system. The resulting "fc-frame expansion is then defined by [258]... 

Establish a global coordinate system, and use a cubic box to confine the model scope. In this case, a point on the west side of the slope is taken as the origin the X-axis and Y-axis point to the east and north, respectively and the Z-axis is perpendicular to the XOY plane. The dimensions of the model are 20 m in the X-axis, 20 m in the Y-axis and 20 m in the Z-axis. 

Build a local coordinate system for each discontinuity set. In each local coordinate system, the origin overlaps with the origin of the global coordinate system, and the Z-axis is parallel to the mean normal vector direction of each set. 

Consider a thin plate of homogeneous, isotropic and linearly elastic material with modulus of elasticity E and Poisson ratio v, having constant thickness hp and occupying the two dimensional multiply connected region of the x,y plane bounded by the piecewise smooth fC-fl curves ro,ri,...,rx-i,rx, as shown in Fig. 1. The plate is stiffened by a set of /= 1,2,..., 7 arbitrarily placed parallel beams of arbitrary doubly symmetric cross section of homogeneous, isotropic and linearly elastic material with modulus of elasticity and Poisson ratio which may have either internal or boundary point supports. For the sake of convenience the x axis is taken parallel to the beams. The stiffened plate is subjected to the lateral load g = g(x, r), x x,y, t>0. For the analysis of the aforementioned problem a global coordinate system Oxy for the analysis of the plate and local coordinate ones O x y corresponding to the centroid axes of each beam are employed as shown in Fig. 1. 

Furthermore, in order to calculate specifically a polymer system we shall define it as being infinite in one dimension and finite in the other two. Therefore, we will consider it as being an infinite, periodic, isolated and helical chain with a straight helical axis (see Fig. 2). In a global coordinate system, we can describe the position of the atom of the nth unit cell by. 

As a simple illustration, consider a thin film of cubic material deposited on a planar substrate surface. Furthermore, suppose that the cube axes coincide with the axes of the global coordinate system. In other words, the material directions [100], [010] and [001] coincide with the xi—axis, the a 2—axis and the 0 3—axis in the global coordinate system. Then Qij = the identity matrix. The mismatch strain is assumed to be equi-biaxial ex-... 

Suppose that a thin film of a cubic material is deposited on a substrate as a single crystal, oriented so that its (111) crystallographic plane is parallel to the substrate surface. The elastic constants of the material are c j- in a coordinate system aligned with the cube axes of the material. A global coordinate system is introduced with the X3—axis normal to the film-substrate interface, as in Section 3.5.1. In this frame, the known components of mismatch strain are = Cm, <... 

Fig. 1.13. The Z-axis of the global coordinate system is along the incident direction and the XZ-plane is the scattering plane...

For the complete uniform distribution function, the external excitation is a vector plane wave propagating along the Z-axis of the global coordinate system and the scattering plane is the XZ-plane. The code computes the following orientation-averaged quantities ... 

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